Competent Reasoning With Rational Numbers

Abstract

Abstract This study, in which I analyzed students' reasoning with rational numbers expressed as fractions, challenges the view that competent numerical reasoning depends on learning a few general methods from instruction and applying them consistently to solve problems. The results indicate that competent reasoning depends on a much richer and more diverse knowledge base that includes many numerically specific and invented strategies, as well as the general strategies learned from instruction. These strategies are richly connected and flexibly applied to solve problems. Elementary school, middle school, and high school students solved four tasks that assessed their knowledge of fraction order and equivalence. The detailed analysis of their solutions revealed four basic perspectives on rational numbers: part-whole relations within divided quantities, relations between numerical components (numerators and denominators), locations relative to well-known reference numbers, and transformations to equivalent numerical forms. Most of the skilled students did not depend on the general transformation strategies emphasized in school (e.g., conversion to common denominator). More often they applied strategies that were specifically tailored to restricted classes of fractions (e.g., fractions with equal numerator) and produced fast and reliable solutions with a minimum of computation effort. A textbook analysis suggested that many were products of student invention. For most of the skilled students, the general transformation strategies were used as a last resort when an appropriate specific strategy could not be found.